Mean Motion Resonances and the Stability of a Circumbinary Disk in a Triple Stellar System

A&A 544, A63 (2012) Astronomy DOI: 10.1051/0004-6361/201219067 & c ESO 2012 Astrophysics

Mean motion resonances and the stability of a circumbinary disk in a triple stellar system

R. C. Domingos, O. C. Winter, and V. Carruba

UNESP, Univ. Estadual Paulista, Grupo de Dinâmica Orbital e Planetologia, Guaratinguetá, SP 12516-410, Brazil e-mail: [rcassia;vcarruba]@feg.unesp.br; [email protected]

Received 17 February 2012 / Accepted 21 June 2012

ABSTRACT

We numerically investigated the orbital stability of a circumbinary disk in a 3-D triple stellar system. We veriﬁed that there is a stable region (protected region) in which highly eccentric and/or inclined orbits can remain stable. In this paper we identify two-body mean motion resonances as a powerful mechanism to increase the eccentricity of the particle on short time-scales (10 kyr or less) and produce a longer-lived high-eccentricity population. These resonances are of high order and have not been previously considered in triple stellar systems. We show that this powerful mechanism can lead to regions containing instabilities and gaps. This process also produces a number of highly eccentric particles whose orbits remain stable. We show that there are limit values of eccentricity and semi-major axis for an orbit to be within the stability region. These quantities represent the dynamical eﬀects of the inner binary and third star companion on a circumbinary disk. The value of this limit depends on the system parameters and should be considered in estimates of the stability, formation, and survival of bodies in triple systems. Key words. celestial mechanics – binaries: close – protoplanetary disks

1. Introduction binary orbital plane. The test particles are initially taken on placed in a cloud of orbits. Verrier & Evans (2008)showedgaps Over the past decade the Spitzer Space Telescope (Werner in the disk and three distinct stable populations of test particles: et al. 2004) has provided complementary information about a prograde disk, a retrograde disk, and a high-inclination halo. known debris disks in multiple-star systems. Observations of pairs of main sequence stars made by Spitzer revealed sta- It is important to study what happens in debris disks to understand the role of stellar companions in planet formation. A ble circumbinary disks around pairs of stars with separations ff of 0.04–5.31 AU in 14 systems (Trilling et al. 2007,seeTable4 stellar companion within 100 AU probably a ects the formation of that paper). In addition, two of these binary systems have an of giant planets (e.g., Nelson 2000; Mayer et al. 2005; Thébault additional companion. et al. 2006; and others). Another example of a debris disk in a multiple star system In this work, we extend the investigations of debris disks per- is the circumbinary disk of HD 98800. This system is a young formed by Akeson et al. (2007) and Verrier & Evans (2008). We (<10 Myr) member of the TW Hydra association (Fekel & Bopp have considered a range of inclinations for the third star and for 1993). HD 98800 has four stars in two binaries, A and B, trav- the particle’s disk. We estimate the stable region in which par- elling around each other in highly inclined and eccentric orbits. ticles can survive despite the third star’s inclination and eccen- According to Furlan et al. (2007), the observations suggests the tricity. Our goal is not to reproduce the current configuration of existence of an optically thick wall at 5.9 AU around the B com- the debris disk of HD 98800, but to sample the range of pos- ponents and an inner, empty region. However, some optically sible outcomes for different initial inclinations of the disk and thin dust orbits the B binary in a ring between 1.5 and 2 AU. the third star. The astronomical motivation for our study is to In particular, Furlan et al. (2007) have argued that the peculiar investigate whether/where the orbital stability region of a debris structure and apparent lack of gas in the HD 98800 disk suggest disk is possible around the close binary, and to show how the that this system is likely already at the debris disk stage. disk structure can be shaped by its dynamical interaction with In later studies of debris disks in triple star systems (Akeson an eccentric and inclined third star. Investigating the details of et al. 2007; Verrier & Evans 2008), the quadruple stellar system these dynamical interactions is beyond the scope of the present HD 98800 was studied as a triple system consisting of the B in- work, therefore we will focus the possible impact of the mean ner binary orbited by a third star. Akeson et al. (2007) investi- motion resonances on the disk because recent studies have men- gated the influence of a third star, with an orbital plane initially tioned them as a powerful mechanism for instabilities and gaps inclined at 10◦ with respect to the B inner binary orbital plane, in the disk, without investigating them in detail (Verrier & Evans on the HD 98800 disk. The authors found that the inclined stel- 2008; Farago & Laskar 2010). An analytical study of mean mo- lar orbit could produce a warp in the disk around the B inner tion resonances in this framework is intended to be carried out binary. Verrier & Evans (2008) considered the third star in three in the future. possible current orbital configurations of the HD 98800 system This paper is structured as follows. In Sect. 2, an overview of (Tokovinin 1999). The third star is considered to be in an or- the empirical expressions of the stability boundary applied here bital plane initially inclined at ∼143◦ with respect to the B inner is presented. In Sect. 3, we present the initial conditions for the Article published by EDP Sciences A63, page 1 of 8 A&A 544, A63 (2012)

Table 1. Orbital elements of the B-A orbit of the HD 98800 system triple system. The empirical expressions for inner and outer bor- (Tokovinin 1999). ders are written as follows: a = 2.92 + 4.21e − 2.67e2 − 1.55μ a (1) Orbital Parameter I II III I B B B B a (UA) 61.9 67.6 78.6 a = 0.477 − 0.412μ − 0.708e + 0.794μ e + 0.276e2 A O A A A A A eA 0.3 0.5 0.6 ◦ − . μ 2 , ΩA ( ) 184.8 184.8 184.8 0 599 AeA aA (2) ◦ ωA ( ) 210.7 224.6 224.0 Table 2 gives the values of aO for each studied configuration. For the inner stability boundary, we found a value of ∼3.87 AU. Another approximation of the stability boundary for plane- Table 2. Outer critical semi-major axis, aO, for the three configurations tary orbits in binary systems can be obtained using the results investigated. from Domingos et al. (2006), who numerically studied the orbital stability of a particle in the framework of the elliptic re- Authors I II III stricted three-body problem. Empirical expressions of the sta- Holman & Wiegert (1999) 10.97 7.96 7.04 bility boundaries for prograde and retrograde orbits were also Domingos et al. (2006) 11.40 8.83 8.08 derived. The authors have found aO values that agree well with Verrier & Evans (2007) 11.15 8.25 7.37 those from Holman & Wiegert (1999) and Verrier & Evans Numerical results 12.20 8.80 7.80 (2007). The results from Domingos et al. (2006)differ from pre- vious works in that the empirical expression for the outer critical Notes. The results are given in Astronomical Unit (AU). semi-major axis of the particle also depends on the eccentricity e of the particle. In the case of a prograde orbit, the outer semi-major axis is given by disk and the triple stellar system assumed here. In Sect. 4, we summarise the relevant results of our numerical simulations and aO = 0.4895(1 − 1.0305eA − 0.2738e)RH, (3) discuss the effects of the inner binary and third star perturba- where RH is the radius of the system’s B-A Hill’s sphere. The tions. In Sect. 5, we present a study on the identification of mean = motion resonances in the particle disk. Finally, our conclusions aO values for e 0 for each configuration of the third star are are presented in Sect. 6. shown in Table 2. Analysing Eqs. (2)and(3), the aO value strongly depends on the eA, aA and μA values when the particle is on circular orbit. 2. Stability boundary This relation may be a result of the fact that the third star’s gravitational perturbation on the particle can change significantly de- To study the stability boundary of a circumbinary disk in a pending on those parameters. This changes the minimum dis- triple stellar system, we have chosen the HD 98800 system. This tance between the third star and the inner binary. The third star’s system has been modelled as a triple stellar system in which gravitational perturbation on the components of the inner binary the B binary is orbited by a particle disk and a more distant third can also change, resulting in a perturbation on the particle. Thus, star. The A pair is treated as a single star disturber. the combination of these two effects must be important to the The B binary orbit has a semi-major axis of aB = 0.983 AU particle’s dynamical stability. According to Eq. (3), the critical and an eccentricity of eB = 0.7849. The solar masses of the semi-major axis aO beyond which the particle would not be sta- BpairareM1 = 0.699 and M2 = 0.582, and the mass ratio ble also depends on the e value. μB = M2/(M1 + M2)is0.45(Bodenetal.2005). The A bi- For the inner stability boundary, the expression from nary (hereafter, third star) has three possible orbital configura- Domingos et al. (2006) does not determine the aI value for the tions (see Table 1), and its mass M3 was assumed to be the sum system considered here. In that study, the aI value represents the of the masses of the B pair (M3 = 1.281 solar masses). The mass semi-major axis of the collision of the particle with the body that ratio μA = M3/(M1 + M2 + M3) is then 0.5 (Tokovinin 1999). it was orbiting. Holman & Wiegert (1999) and Verrier & Evans To study the stability of planets in binary star systems, (2007)haveshownthataI depends on eB because it corresponds Holman & Wiegert (1999) have derived empirical expressions to the semi-major axis related to the centre of mass of the inner of stability boundaries for planetary orbits in binary star sys- binary, from which the orbit of the particle can become unstable. tems. According to their results, the stability region is delimited However, the eccentricity of the particle was not considered in by what we called internal (aI) and outer (aO) critical semi-major those studies. axes. The internal critical semi-major axis is the closest stable Considering that particles can be in high-e orbits with a peri- orbit to the two stars (or one of the stars) due to the star pair’s centre distance of Rp = a(1−e) within the unstable region, parti- circumbinary perturbations. The outer critical semi-major axis cles might be removed by perturbations from the inner binary. To is the outermost stable orbit around the star pair (or one of the estimate the value of e as a function of the minimum semi-major stars) due to the perturbations of a distant star. axis a, in which the particle could move near the inner binary, Using the approximated expressions from Holman & we have assumed that Rp is given by Eq. (1). The eccentricity of Wiegert (1999), we give the values of aO for each orbital con- the particle as a function of a is then given by figuration of the third star in Table 2. We found an inner stability e = 1 − (R /a), (4) boundary of ∼4.04 AU. p Verrier & Evans (2007) have performed numerical simula- we notice that the difference between the aI values from Holman tions on the stability zones for particles orbiting an inner binary &Wiegert(1999) and those from Verrier & Evans (2007)were in the presence of a more distant third star. They have presented not significant. The advantage of our approach is that the aI value empirical expressions of the stability boundary for coplanar cir- depends on not only μB, aB and eB, but also on a whole set of cumbinary stability by modelling the stars as a low-inclination particle eccentricities.

A63, page 2 of 8 R. C. Domingos et al.: Mean motion in a circumbinary disk

Assuming a limit of eccentricity, elim, for an orbit with a libration, they are destabilised and ejected from the system or semi-major axis limit, alim, to be within the stable region, it collide with the inner binary. was possible to establish the eccentricity from expressions (3) and (4). The (alim, elim) maximum pair was estimated by setting Rp equal to the outer stability semi-major axis, aO.For 3. Numerical simulations the I, II and III configurations, these values are approximately Using the three possible fits for the third star given in Table 1, (8.830 AU, 0.572), (6.670 AU, 0.433) and (5.963 AU, 0.366), we numerically investigated the stability of the particles around respectively. For the a = a case with e exceeding e ,the lim lim the B pair. The B binary orbit was assumed to be in the plane. pericentre and/or apocentre distances can be located outside the The orbit of the third star was placed around the centre of mass stable region. When e = e , the particle orbits have peri- lim of the B pair. The initial inclinations for the third star (I )are centre and apocentre distances on the inner and outer stability A given with respect to the initial B binary plane and are taken to boundaries, respectively. ◦ ◦ ◦ be from 0 to 50 with ΔIA = 10 . According to the goal of this work, we focused on particles The disk was formed by 30 000 massless particles (planetes- within the stable region. In our analyses, we used the definition imals) distributed on the same plane around the B binary with of the (alim, elim) pair as a reference value. These physical quan- semi-major axis (a) from 2 to 30 AU. The initial inclination (i)of ff ◦ ◦ ◦ tities represent the dynamical e ects of the inner binary and thediskwasassumedtovaryfrom0 to 90 with Δi = 10 third star companion on a circumbinary disk and consequently with respect to the initial B binary plane. The initial eccentric- − represent the particle stability. The limit radius is between aI ities (e) of the particles were randomly distributed within 10 3. and aO values. The combination of elim and alim represents a Longitudes of pericentre and mean longitudes were distributed critical value of high eccentricity and semi-major axis values of randomly between 0◦ and 360◦. Our reference plane is the B or- orbits relative to the particle instabilities. A specific (a, e)pair bital plane. In particular, our main interest is the final outcome can result in an orbit of collision with the inner binary or escape of the evolution of the disk with respect to the initial B orbital out of the system. plane at time t = 0. The numerical simulations were stopped after 1 Myr and interrupted if any of the following situations occurred: (i) the 2.1. Critical inclination planetesimal travelled closer than 0.01 AU from B binary com- It is particularly interesting to determine the conditions under ponents or (ii) the planetesimal travelled farther than 200 AU which an inclined protoplanetary disk remains stable. The mean from B binary (ejection of the system). The particles surviving inclination i of a disk might also have implications for the sta- the full integration time were considered stable. We used the bility region of a system in which large bodies disturb the disk, Swift package (Wisdom & Holman 1991; Levison & Duncan producing dust through collisional events. Depending on the sys- 1994) and the Hierarchical Jacobi Symplectic algorithm (Beust tem characteristics, something will disturb the disk’s increasing 2003). i value (Artymowicz 1997; Thébault & Augereau 2007; Quillen et al. 2007). An important question regarding this problem is the existence of a critical value for the inclination (∼39.23◦) 4. Results and discussion between the orbital planes of the disturbed and disturbing bod- Our numerical results are summarised in Figs. 1 to 4,whichre- ies (restricted three-body problem). If the disturbing body is in fer to the final orbital evolution of the particles on the a − e a highly inclined orbit relative to the orbital plane of the dis- and a − i planes. Similar results have been obtained for three turbed body, the eccentricity increases and the near-circular or- orbital configurations. As an example, Figs. 2 to 4 show the final bit becomes highly elliptic. This effect is called the Kozai-Lidov results for the orbital configuration I. mechanism (Kozai 1962)and(Lidov1962). The Kozai-Lidov mechanism causes strong periodic variations of the particle eccentricity and inclination while conserving 4.1. The coplanar case their semi-major axes. The concurrent increase in the inclination Figure 1 shows the final results of the numerical simulations for and decrease in the eccentricity usually leads to instability in the orbital configurations I, II and III for the coplanar regime (IA = system. Assuming that the particle’s initial eccentricity is close i = 0). In configuration I, the outer critical semi-major axis is to zero, we can obtain the maximum eccentricity of a particle much larger than that in configurations II and III. Figure 1 shows given an initial inclination (Innanen et al. 1997). Carruba et al. the final eccentricity as a function of the final semi-major axis of (2002) have shown that particles on higher inclination librating the particles. The stable region is delimited by the two borders, in the Kozai-Lidov resonance may not reach the maximum ec- aI and aO. The left border (red curve) and the right border (green centricity and can be stable for millions of years. curve) are given by Eqs. (4)and(3). Table 2 gives the values In the system studied here, the Kozai-Lidov mechanism of aO obtained for the simulations for each orbital configuration. takes effect on the circumbinary disk when the relative incli- The inner critical semi-major axis at and beyond which particles nation of the particle and third star orbital planes is greater survive the integration is ∼3.3 AU. than 40◦. Verrier & Evans (2009) showed that the inner bi- In Fig. 1, it can be noted that particle orbits were not re- nary may cause a nodal libration instead of Kozai-Lidov cycles, stricted to within a (a, e) triangular region. Orbits within this which stabilises the test particles against any Kozai-Lidov insta- region are considered to be stable, and orbits located outside bility driven by the outer star. These authors also reported that this region are considered to be unstable. It is important to note particles closer to the inner binary have nodal libration periods that the simulations were halted after 1 Myr, which corresponds shorter than Kozai-Lidov cycles. Therefore, the nodal libration to 10% of the age of a system such as HD 98800. If the simu- dominates, resulting in the stability of the particle. Otherwise, lations were continued, some of the remaining particles whose when particles more distant (closer to the outer border of the sta- orbits were outside of the stability region would have also been bility region) have Kozai-Lidov periods shorter than the nodal ejected or collided.

A63, page 3 of 8 A&A 544, A63 (2012)

1 limit must be shorter than that for systems with low eccentric- I ities of the third star. The three estimates of the value of elim 0.8 aO are in reasonable agreement with the results of our numerical e simulations. 0.6 lim aI e 0.4 4.2. Stellar perturbations on inclined orbits

0.2 In this section, we present results for inclined orbits of the third star and the particle disk. First, we have considered the case 0 2 3 4 5 6 7 8 9 10 11 12 13 of an inclination between the B pair orbit and the particle’s a (AU) disk, followed by an inclination between the disk and the third star orbit. 1 II 0.8 aO 4.2.1. Inclined particle disk

0.6 Figure 2 shows that for the initial inclinations of 10◦ and 20◦ the

e e lim final results in a − e planes are identical to those shown in Fig. 1. 0.4 a The final distribution of the eccentricity of the particles within I ff 0.2 the “protected region” also does not significantly di er from those in Fig. 1. The particle concentrations in high-eccentricity 0 orbits have not changed location with respect to the coplanar 2 3 4 5 6 7 8 9 case. Figure 2 shows the final inclination of the particles. For the ◦ a (AU) disk initially inclined by 20 the distribution of maximum final inclination of the particles was ∼60◦. Despite a higher final incli- 1 III nation, the final distribution of the number of particles is similar 0.8 to that for the coplanar case. In particular, for configuration I, we note that the particle concentration for the higher inclination a 0.6 O occurs at a = 4.5AU. aI

e Figure 3 shows the final orbital eccentricities and inclina- e ◦ ◦ 0.4 lim tions of the particles for a disk initially inclined by 30 ,50 and 90◦. Evidently, a significant number of particles have their 0.2 eccentricities and inclinations excited to very high values for the disk initially inclined by 30◦. We have seen that many particles 0 were quickly ejected in the vicinity of the outer boundary. In 2 3 4 5 6 7 8 9 the inclination plot, we can note that between ∼4.0 and 7.5 AU a (AU) particles are excited in inclination up to ∼160◦. ◦ ◦ Fig. 1. Region of stability of the particles in the space of a versus e for For particle disks with initial inclinations 50 and 90 ,re- the coplanar case (IA = i = 0). The plots represent the final results for sults showed that some particles remained stable for high val- the orbital configurations I, II and III. The red and green curves give the ues of initial inclination. Note that these particles have sig- analytical estimates for the stability region (“protected region”). These nificantly increased their eccentricities, but the eccentricities curves represent the internal, aI and outer, aO, critical semi-major axes. did not reach sufficiently high values to be removed from the The red and green curves are given by Eqs. (4)and(3), respectively. system during the integration time. On the other hand, it can The dashed curve represent the (alim, elim) pair for a particle within the be noticed that many particles have not significantly increased “protected region”. eccentricities. We can also note that for i = 90◦ particles are stable up to a ∼< 9.0 AU, with gaps appearing around ∼2.5 AU and 3.1 AU. The surviving particles within the stable region (hereafter the In general, most of these particles have final eccentricities of e ≤ “protected region”) are expected not to be ejected or even col- 0.1 when the disk orbit is inclined. Depending on the initial value lide with the central stars. All particles were initiated with near of i,theaI value at and beyond which particles survive the in- circular orbits; at some point, some particles achieved eccentric tegration is within 2.4 to 3.3 AU. The lower value for aI was orbits. For orbital configuration I, some particles are excited to obtained by considering high relative inclinations (i > 50◦). higher eccentricities around a = 4.0, 4.2, 10.0, 10.5 and 11.5 AU. For orbital configurations II and III, particles in high-e orbits appear around 5.0 AU and 5.5 AU, respectively, and appear close 4.2.2. Relative inclination between the particle disk to the outer border. In general, when particles are not in high-e and third star orbits, they remain in orbits with e ≤ 0.1 within the “protected region”. Another perspective is to initially consider the particle disk and Our results have indicated high sensitivity to the third star’s third star on non-coplanar orbits. Mazeh & Shaham (1979)have eccentricity. For configurations with a high eccentricity of the shown that when the inclination of the third star is nonzero, the third star (0.5 and 0.6), particles in the outer region of the eccentricity of the inner binary oscillates (with corresponding disk (a ≥ 8.0 AU) are ejected from the system on a short oscillations in its inclination), while IA and eA remain approx- timescale. For large eccentricities of the third star, the value imately constant. As expected, our numerical simulations have ◦ of elim decreases, and the timescale for particles reaching this shown that for higher values of IA (40 ), the effect of the inner

A63, page 4 of 8 R. C. Domingos et al.: Mean motion in a circumbinary disk

200 200 10o 20o 160 160

120 120 ree) ree) g g 80 80 i (de i (de

40 40

0 0 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 10 11 12 13 a (AU) a (AU) Fig. 2. Regions of stability of an inclined particle disk for I orbital conﬁguration. Each plot shows the ﬁnal results of the particles in a − i planes. The initial inclination of the disk is indicated on the upper right corner of each panel.

1 200 30o 30o 0.8 aO 160

0.6 120

a ree)

I g e 0.4 80 i (de

0.2 40

0 0 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 10 11 12 13 a (AU) a (AU) 1 200 50o 50o 0.8 aO 160

0.6 120

a ree)

I g e 0.4 80 i (de

0.2 40

0 0 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 10 11 12 13 a (AU) a (AU) 1 200 90o 90o 0.8 aO 160

0.6 120

a ree)

I g e 0.4 80 i (de

0.2 40

0 0 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 10 11 12 13 a (AU) a (AU) Fig. 3. Plots of a − e and a − i for particle disks initially inclined by 30◦,50◦ and 90◦.

binary perturbations on the inner edge of the particle disk be- distance to the binary at which particles survived was 2.4 AU. As comes important in determining the inner stability boundary. shown in the figure, some results revealed gaps (around 4.5 AU Figure 4 presents a representative sample of the final re- and 5.5 AU) and a diffusion of the particles, with subsequent sults when the particle disk and third star were placed in non- particle ejection or collision with the inner binary. ◦ coplanar orbits. The initial IA value was 30 , and the initial in- In general, we found three distinct and clearly located struc- clinations of the particle disk were 20◦,50◦ and 90◦ (indicated in tures: (i) a chaotic region closer to the inner binary, (ii) a stable the upper right corner of the plot). Varying i and IA, the closest region where particles on highly inclined orbits can survive, and

A63, page 5 of 8 A&A 544, A63 (2012)

1 200 20o 20o 0.8 aO 160

0.6 120

a ree)

I g e 0.4 80 i (de

0.2 40

0 0 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 10 11 12 13 a (AU) a (AU) 1 200 50o 50o 0.8 aO 160

0.6 120

a ree)

I g e 0.4 80 i (de

0.2 40

0 0 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 10 11 12 13 a (AU) a (AU) 1 200 90o 90o 0.8 aO 160

0.6 120

a ree)

I g e 0.4 80 i (de

0.2 40

0 0 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 10 11 12 13 a (AU) a (AU) Fig. 4. Diagrams of a − e and a − i. The initial inclination of the particle disk was indicated in the upper right corner of the panel and is relative to ◦ the B binary orbital plane. The results shown are for the case IA = 30 .

(iii) an unstable region that depends on the distance of the third longitudes that correspond to the particle, the inner binary (Ba star and the relative inclination of the particles. Depending on and Bb stars), and the third star, respectively, and 1 is the peri- initial conditions from the third star and the particle disk, eccen- centre longitude of the particle. A particle is involved in a mean tricity peaks and gaps can appear within the stable region. motion resonance when the critical argument rate corresponds to j1λ˙ 1 + j2λ˙ 2 + j3λ˙3 + j4λ˙ 4 + j5˙ 1 ∼ 0, where λ˙ 2 and λ˙ 3 denote the mean motions of the inner binary, λ˙ 1 and λ˙ 4 are the mean 5. Mean motion resonances motions of the particle and the third star, respectively, and ˙ 1 is Our investigation was motivated by the eccentricity peaks shown the pericentre rate of the particle. by some particles with low inclination (i ≤ 30◦), which might It is well known (see for instance Murray & Dermott 1999) be due to resonant effects. Thus, instabilities and gaps appear that the orbital periods of each star of the B pair about its centre of mass are the same. The mean longitudes in their orbits dif- around these resonances as a result of high eccentricities. ◦ We are studying a system with four bodies, none of which fer by 180 . Therefore, there is only one frequency (the mean motion) associated with the B pair, rather than two. can be considered a central body. Thus, there are four natural λ˙ = λ˙ = λ˙ φ˙ frequencies in the system given by the mean motions of each Taking 2 3 B, we rewrite in the following form: body. Given that the stars have highly eccentric orbits and that φ˙ = j1λ˙ 1 + jBλ˙ B + j4λ˙ 4 + j5˙ 1, (6) this section is restricted to the planar case, the resonant angle involves not only mean motions but also the pericentre longitude in Fig. 1, we can see peaks of eccentricity in regions close to ≤ ≤ of the particle. The expression of the resonant angle associated the inner binary (3 AU a 5 AU) and in regions farther away ≤ ≤ α with the resonance is then given by (9 AU a 12 AU). Let us call the first region and the second region β. α ff φ = j1λ1 + j2λ2 + j3λ3 + j4λ4 + j51, (5) In region , the gravitational e ects of the third star do not destabilise the particle. The third star orbital frequency (λ˙ 4)is where j’s are integers with j1 + j2 + j3 + j4 + j5 = 0 according to much lower than the inner binary orbital frequency (λ˙ B), and the first d’Alembert rule. The λ1, λ2, λ3 and λ4 are mean motion thus contributes very little to the combination given by Eq. (6).

A63, page 6 of 8 R. C. Domingos et al.: Mean motion in a circumbinary disk

Table 3. Mean motion resonances. from 0.08 to 0.23 and the value of a remains close to the resonant value with increasingly erratic oscillations up to approximately

Resonant argument ares (AU) 3240 years, when the particle escapes the system. Thus, we can say that the dynamical behaviour of this particle is governed by 7λ1 − 1λB − 61 3.5971 8λ − 1λ − 7 3.8665 the 8:−1 two-body mean motion resonance. To verify the impor- 1 B 1 9λ1 − 1λB − 81 4.2531 tance of the frequency ˙ 1 in the resonant angle, we plotted the 10λ1 − 1λB − 91 4.5626 resonant angle using only λ˙ 1 and λ˙ B (setting j5 = 0). We verified 11λ1 − 1λB − 101 4.8620 that the plots are almost identical. Therefore, the contribution − λ + λ − 1 1 11 4 10 1 9.9331 of ˙ to the resonant angle is not significant. − λ + λ − 1 1 1 10 4 9 1 10.5847 The P2 particle is an example of a particle near the outer −1λ1 + 9λ4 − 81 11.3549 − λ + λ − border of the stable region. In this resonance, the particle sees 1 1 8 4 7 1 12.2825 the inner binary essentially as a point mass. We show our results for 6000 years to provide a better graphic representation of the resonant angle, but the particle survives the full integration 0.4 time. The resonant angle is φ = −λ1 + 11λ4 − 101,andtheres- onant semi-major axis is 9.9331 AU. According to the results, 0.3 there is no obvious pattern to the libration of the resonant angle. The particles remain oscillating for all times among several values of the resonant angle, and there is no tendency for a circu- e 0.2 lating regime. The semi-major axis and eccentricity oscillations exhibit a good correlation with each other. To verify the impor- 0.1 tance of ˙ 1 on the resonant angle of P2, we plotted the resonant angle using only λ˙ 1 and λ˙ 4 (setting j5 = 0). We verified that the plots are quite different. Therefore, the contribution of ˙ has a 0 1 2 3 4 5 6 7 8 9 10 11 12 13 significant effect on the resonant angle and cannot be neglected. a (AU) In our study of mean motion resonances, we concentrated on the planar case. The resonant angle was defined using only the Fig. 5. Region of stability of the particles in the space of a versus e for mean and pericentre longitudes. Certainly, when the disk and/or the coplanar case (IA = i = 0). The plot represents the final result for the orbital configuration I. The red vertical lines correspond to the locations the third star is inclined, a study considering the node and peri- of the resonances listed in Table 3. centre longitudes would be required to gain a better understanding of the resonant effects.

In practice, we can consider j4 = 0 such that (6) becomes 6. Final remarks φ˙ = j1λ˙ 1 + jBλ˙ B + j5˙ 1. This equation involves two-body resonances because we only used two mean motion frequencies, but In this work, we obtained the borders of the stable region for a physically we have three bodies involved in the resonance with particle disk around a close binary system disturbed by a gravi- λ = λ + ◦ Bb Ba 180 . tational field of a distant (minimum pericentre distance ∼<43 AU) β In contrast, the third star plays an important role in region . companion star. These interactions again involve two-body resonances, with the Our numerical results showed that the orbital evolution of particle seeing the inner binary essentially as a single body. Now the particle disk has been affected by the nodal libration due to = φ˙ = λ˙ + we can consider jB 0 such that (6) becomes j1 1 the inner binary, the Kozai-Lidov effect and several mean motion λ˙ + j4 4 j5 ˙ 1. resonances but is not dominated by any one of these factors in Table 3 presents a list of the two-body mean motion reso- particular. In reality, the situation is much more complex. There nances obtained here. In the first and second columns, we show are several mechanisms, overlaps of mean motion and/or secu- the resonant argument and resonant semi-major axis of the par- lar resonances that increase the particle eccentricities, inducing ticle, respectively. Figure 5 reproduces Fig. 1 with the loca- instability. We have shown the following: tions of these two-body mean motion resonances. In Fig. 6,we show examples of temporal evolutions of a, e,andφ for two 1. There is an eccentricity limit for a particle to remain in the particles (P1 and P2) identified in the libration regime of the stability region. This limit depends on the a value of the par- resonant angle. ticle. We recall that the value of the particle eccentricity can The particle P1 is an example of a particle quite close to the vary significantly due to the resonant effects (mean motion inner binary. We note that the combination of the resonant an- and secular resonances). For particles at high inclination, the gle φ = 8λ1 − λB − 71 (second plot) shows that the particle orbits can be destabilised by the Kozai-Lidov mechanism. is in the 8:−1 two-body mean motion resonance with the inner Depending on the initial conditions, eccentricity and inclina- binary. The time variations of the resonant angle show that it ini- tion peaks appeared in regions where the effects of the nodal tially librates for approximately 3240 years, at which point the libration influence the stability of the particle orbit. We also behaviour of φ, a and e clearly indicate the escape of the parti- showed that this eccentricity limit depends on the system pa- cle. An important point to note is the manner in which the res- rameters and should be considered when estimating the sta- onance appears on the plot. There is no obvious pattern to the bility of orbits in triple systems. libration of the resonant angle, but this fact alone does not mean 2. Among the mechanisms that can increase particle eccentric- that the orbit is chaotic. The semi-major axis and eccentricity os- ity, we have identified two-body mean motion resonances as cillations exhibit good correlation. Note that the variations in a a powerful mechanism to increase the particle eccentricity on and e are not different. After approximately 1900 years, the evo- short time-scales (10 kyr or less) and produce a longer-lived lution of the eccentricity undergoes stronger irregular variations high-eccentricity population. These resonances are of high

A63, page 7 of 8 A&A 544, A63 (2012)

4.5 10.5 P1 P2 4.3 10.3

4.1 10.1 a (AU) 3.9 a (AU) 9.9

3.7 9.7

3.5 9.5 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 6000 Time (years) Time (years) 360 360 )

300 g 300 ) g (de 1

(de 240 240 ϖ 1 ϖ

180 − 10 180 −7 4 B λ λ −

1 120 120 +11 λ 1 8 λ

60 − 60

0 0 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 6000 Time (years) Time (years) 0.5 0.5

0.4 0.4

0.3 0.3 e e 0.2 0.2

0.1 0.1

0 0 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 6000 Time (years) Time (years) Fig. 6. Orbital evolutions of the semi-major axis, resonant angles, and eccentricity as functions of time. At the top of the ﬁrst plot, we indicate the corresponding particle. The results in the ﬁrst column are for a particle with initial conditions a = 3.8398 AU and e = 0.00070. The second column is for a particle with initial conditions a = 10.0980 AU and e = 0.00049. Note that the evolution of φ (second plot) shows that the P1 and P2 particles cross the two-body mean motion resonance and that the libration of the φ angle is evidence of an association between the orbits and the resonances.

order and are considered a weak class of resonances in the Carruba, V., Burns, J. A., Nicholson, P. D., & Gladman, B. J. 2002, Icarus, 158, outer asteroid belt; thus, they have not previously been con- 434 sidered in stellar systems. A study of the dynamic behaviour Domingos, R. C., Winter, O. C., & Yokoyama, T. 2006, MNRAS, 373, 1227 Farago, F., & Laskar, J. 2010, MNRAS, 401, 1189 of particles in mean motion resonances is in progress and Fekel, F. C., & Bopp, B. W. 1993, ApJ, 419, L89 will be the subject of a future paper. Furlan, E., Sargent, B., Calvet, N., et al. 2007, AJ, 664, 1176 Holman M. J., & Wiegert P. A. 1999, AJ, 117, 621 Although we have used the specific parameters for HD 98800, Innanen K. A., Zheng, J. Q., Mikkola, S., & Valtonen, M. 1997, AJ, 113, 1915 we hope that our results can contribute for understanding how Kozai, Y. 1962, AJ, 67, 591 stellar companions can affect the evolution of disks around close Levison, H. F., & Duncan, M. J. 1994, Icarus, 108, 18 binaries for any system. Lidov, M. L. 1962, P&SS, 9, 719 Mayer, L., Wadsley, J., Quinn, T., & Stadel, J. 2005, MNRAS, 363, 641 Mazeh, T., & Shaham, J. 1979, A&A, 77, 145 Acknowledgements. The authors thank Hervé Beust for providing his HJS Murray, C. D., & Dermott, S. F. 1999, Solar System Dynamics (Cambridge, algoritm and James Freddy L. Machuca for helpful discussions and com- England: Cambridge University Press) ments on resonant dynamics. This work had financial support from UNESP, Nelson, A. F. 2000, ApJ, 537, L65 CNPq and FAPESP (grant number 2008/08679-4). These supports are gratefully Quillen, A. C., Morbidelli, A., & Moore, A. 2007, MNRAS, 380, 1642 acknowledged. Thébault, P., & Augereau, J. C. 2007, A&A, 472, 169 Thébault, P., Marzari, F., & Scholl, H. 2006, Icarus, 183, 193 Tokovinin, A. 1999, Astron. Lett., 25, 669 Trilling, D. E., Stansberry, J. A, Stapelfeldt, K. R., et al. 2007, ApJ, 658, 1289 References Verrier, P. E., & Evans, N. W. 2007, MNRAS, 382, 1432 Verrier, P. E., & Evans, N. W. 2008, MNRAS, 390, 1377 Akeson, R. L., Rice, W. K. M., Boden, A. F., et al. 2007, ApJ, 670, 1240 Verrier, P. E., & Evans, N. W. 2009, MNRAS, 394, 1721 Artymowicz, P. 1997, Ann. Rev. Earth Planet Sci., 25, 175 Werner, M. W., Roelling, T. L., Low, F. J., et al. 2004, Ap&SS, 154, 1 Beust, H. 2003, A&A, 440, 1129 Wisdom, J., & Holman, M. 1991, AJ, 102, 1528 Boden, A. F., Sargent, A. I., akeson, R. L., et al. 2005, AJ, 635, 442

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